The Smarandache vertices of comaximal graph of a commutative ring

Abstract

Let R be a commutative ring with identity 1 ̸= 0. Define the comaximal
graph of R, denoted by CG(R), to be the graph whose vertices are the elements of R,
where two distinct vertices a and b are adjacent if and only if Ra + Rb = R. A vertex a
in a simple graph G is said to be a Smarandache vertex (or S-vertex for short) provided
that there exist three distinct vertices x, y, and b (all different from a) in G such that
a—x, a—b, and b—y are edges in G but there is no edge between x and y. The main
object of this paper is to study the S-vertices of CG(R) and CG2(R) \ J(R) (or CGJ (R)
for short), where CG2(R) is the subgraph of CG(R) which consists of nonunit elements
of R and J(R) is the Jacobson radical of R. There is also a discussion on a relationship
between the diameter and S-vertices of CGJ (R).